TheInfoList

In
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, an abstract mathematical discipline, a nodal decomposition of a morphism $\varphi:X\to Y$ is a representation of $\varphi$ as a product $\varphi=\sigma\circ\beta\circ\pi$, where $\pi$ is a strong epimorphism, $\beta$ a bimorphism, and $\sigma$ a strong monomorphism.A
monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
$\mu:C\to D$ is said to be strong, if for any
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
$\varepsilon:A\to B$ and for any morphisms $\alpha:A\to C$ and $\beta:B\to D$ such that $\beta\circ\varepsilon=\mu\circ\alpha$ there exists a morphism $\delta:B\to C$, such that $\delta\circ\varepsilon=\alpha$ and $\mu\circ\delta=\beta$

# Uniqueness and notations

If exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions $\varphi=\sigma\circ\beta\circ\pi$ and $\varphi=\sigma\text{'}\circ\beta\text{'}\circ\pi\text{'}$ there exist
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s $\eta$ and $\theta$ such that : $\pi\text{'}=\eta\circ\pi,$ : $\beta=\theta\circ\beta\text{'}\circ\eta,$ : $\sigma\text{'}=\sigma\circ\theta.$ This property justifies some special notations for the elements of the nodal decomposition: :$\begin & \pi=\operatorname_\infty \varphi, && P=\operatorname_\infty \varphi,\\ & \beta=\operatorname_\infty \varphi, && \\ & \sigma=\operatorname_\infty \varphi, && Q=\operatorname_\infty \varphi, \end$ – here $\operatorname_\infty \varphi$ and $\operatorname_\infty \varphi$ are called the ''nodal coimage of $\varphi$'', $\operatorname_\infty \varphi$ and $\operatorname_\infty \varphi$ the ''nodal image of $\varphi$'', and $\operatorname_\infty \varphi$ the ''nodal reduced part of $\varphi$''. In these notations the nodal decomposition takes the form :$\varphi=\operatorname_\infty \varphi\circ\operatorname_\infty \varphi \circ \operatorname_\infty \varphi.$

# Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category  each morphism $\varphi$ has a standard decomposition : $\varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi$, called the ''basic decomposition'' (here $\operatorname \varphi=\ker\left(\operatorname \varphi\right)$, $\operatorname \varphi=\operatorname\left(\ker\varphi\right)$, and $\operatorname \varphi$ are respectively the image, the coimage and the reduced part of the morphism $\varphi$). If a morphism $\varphi$ in a pre-abelian category  has a nodal decomposition, then there exist morphisms $\eta$ and $\theta$ which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities: : $\operatorname_\infty \varphi=\eta\circ\operatorname \varphi,$ : $\operatorname \varphi=\theta\circ\operatorname_\infty \varphi\circ\eta,$ : $\operatorname_\infty \varphi=\operatorname \varphi\circ\theta.$

# Categories with nodal decomposition

A category  is called a ''category with nodal decomposition'' if each morphism $\varphi$ has a nodal decomposition in . This property plays an important role in constructing envelope (category theory), envelopes and refinement (category theory), refinements in . In an abelian category  the basic decomposition : $\varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi$ is always nodal. As a corollary, ''all abelian categories have nodal decomposition''. ''If a pre-abelian category  is linearly complete, A category  is said to be ''linearly complete'', if any functor from a linearly ordered set into  has direct limit, direct and inverse limits. well-powered in strong monomorphismsA category  is said to be ''well-powered in strong monomorphisms'', if for each object $X$ the category $\operatorname\left(X\right)$ of all strong monomorphisms into $X$ is skeletally small (i.e. has a skeleton which is a set). and co-well-powered in strong epimorphisms,A category  is said to be ''co-well-powered in strong epimorphisms'', if for each object $X$ the category $\operatorname\left(X\right)$ of all strong epimorphisms from $X$ is skeletally small (i.e. has a skeleton which is a set). then  has nodal decomposition.'' More generally, ''suppose a category  is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphismsIt is said that ''strong epimorphisms discern monomorphisms'' in a category , if each morphism $\mu$, which is not a monomorphism, can be represented as a composition $\mu=\mu\text{'}\circ\varepsilon$, where $\varepsilon$ is a strong epimorphism which is not an isomorphism. in , and, dually, strong monomorphisms discern epimorphismsIt is said that ''strong monomorphisms discern epimorphisms'' in a category , if each morphism $\varepsilon$, which is not an epimorphism, can be represented as a composition $\varepsilon=\mu\circ\varepsilon\text{'}$, where $\mu$ is a strong monomorphism which is not an isomorphism. in , then  has nodal decomposition.'' The category Ste of stereotype spaces (being non-abelian) has nodal decomposition, as well as the (non-additive category, additive) category SteAlg of stereotype algebras .

# References

* * *{{cite journal, last=Akbarov, first=S.S., title=Envelopes and refinements in categories, with applications to functional analysis, url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513, journal=Dissertationes Mathematicae, year=2016, volume=513, pages=1–188, arxiv=1110.2013, doi=10.4064/dm702-12-2015, s2cid=118895911 Category theory